Perform the row operation, $R_2+3R_1\rightarrow R_2$, on the following matrix. $\left[\begin{array} {ccc} -5 & -1 & 0 & -1 \\ 2 & -3 & 4 & 7 \\ 3 & 3 & 1 & 9 \end{array} \right] $
Answer: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_1$ and $R_2$ are given below. $R_1=\left[\begin{array} {ccc} -5 & -1 & 0 & -1 \end{array} \right] ~~~~~ R_2=\left[\begin{array} {ccc} 2 & -3 & 4 & 7 \end{array} \right]$ We are asked to perform the row operation, $R_2+3R_1\rightarrow R_2$. Therefore, we must add $3R_1$ to $R_2$. $\begin{aligned}R_2+3R_1 &= \left[\begin{array} {ccc} 2 & -3 & 4 & 7 \end{array} \right] + 3\left[\begin{array} {ccc} -5 & -1 & 0 & -1 \end{array} \right] \\\\&=\left[\begin{array} {ccc} -13 & -6 & 4 & 4 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_2$ with $R_2+3R_1$. $\left[\begin{array} {ccc} -5 & -1 & 0 & -1 \\ {2} & {-3} & {4} & {7} \\ 3 & 3 & 1 & 9 \end{array} \right]\xrightarrow{R_2+3R_1\rightarrow R_2} \left[\begin{array} {ccc} -5 & -1 & 0 & -1 \\ {-13} & {-6} & {4} & {4} \\ 3 & 3 & 1 & 9 \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -5 & -1 & 0 & -1 \\ -13 & -6 & 4 & 4 \\ 3 & 3 & 1 & 9 \end{array} \right]$